Optimal matching forests and valuated delta-matroids

Abstract

The matching forest problem in mixed graphs is a common generalization of the matching problem in undirected graphs and the branching problem in directed graphs. Giles presented an O(n2m)-time algorithm for finding a maximum-weight matching forest, where n is the number of vertices and m is that of edges, and a linear system describing the matching forest polytope. Later, Schrijver proved total dual integrality of the linear system. In the present paper, we reveal another nice property of matching forests: the degree sequences of the matching forests in any mixed graph form a delta-matroid, and the weighted matching forests induce a valuated deltamatroid. We remark that the delta-matroid is not necessarily even, and the valuated delta-matroid induced by weighted matching forests slightly generalizes the well-known notion of Dress andWenzel s valuated delta-matroids. By focusing on the delta-matroid structure and reviewing Giles algorithm, we design a simpler O(n2m)-time algorithm for the weighted matching forest problem. By incorporating Gabow s method for the weighted matching problem into Giles algorithm, we also present a faster algorithm for the weighted matching forest problem running in O(n 3)-time, which improves upon the previous best complexity of O(n 2m). © 2014 Society for Industrial and Applied Mathematics.